Conditional Lie Bäcklund Symmetries and Sign-Invariants to Quasi-Linear Diffusion Equations
Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term ut=[um(ux)n]x+P(u)ux+Q(u) , where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic =uxx+H(u)u2x+G(u)(ux)2−n+F(u)u1−nx and the Hamilton–Jacobi sign-invariant J=ut+A(u)un+1x+B(u)ux+C(u) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.
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Document Type: Research Article
Affiliations: Northwest University, He'nan Agricultural University
Publication date: November 1, 2007